Dimension doubling theorem

In probability theory, the dimension doubling theorems are two results about the Hausdorff dimension of an image of a Brownian motion. In their core both statements say, that the dimension of a set $$A$$ under a Brownian motion doubles almost surely.

The first result is due to Henry P. McKean jr and hence called McKean's theorem (1955). The second theorem is a refinement of McKean's result and called Kaufman's theorem (1969) since it was proven by Robert Kaufman.

Dimension doubling theorems
For a $$d$$-dimensional Brownian motion $$W(t)$$ and a set $$A\subset [0,\infty)$$ we define the image of $$A$$ under $$W$$, i.e.
 * $$W(A):=\{W(t): t\in A\}\subset \R^d.$$

McKean's theorem
Let $$W(t)$$ be a Brownian motion in dimension $$d\geq 2$$. Let $$A\subset [0,\infty)$$, then
 * $$\dim W(A)=2\dim A$$

$$P$$-almost surely.

Kaufman's theorem
Let $$W(t)$$ be a Brownian motion in dimension $$d\geq 2$$. Then $$P$$-almost surely, for any set $$A\subset [0,\infty)$$, we have
 * $$\dim W(A)=2\dim A.$$

Difference of the theorems
The difference of the theorems is the following: in McKean's result the $$P$$-null sets, where the statement is not true, depends on the choice of $$A$$. Kaufman's result on the other hand is true for all choices of $$A$$ simultaneously. This means Kaufman's theorem can also be applied to random sets $$A$$.